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19
On the Selfsimilar Nature of Ethernet Traffic (Extended Version)
, 1994
"... We demonstrate that Ethernet LAN traffic is statistically selfsimilar, that none of the commonly used traffic models is able to capture this fractallike behavior, that such behavior has serious implications for the design, control, and analysis of highspeed, cellbased networks, and that aggrega ..."
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Cited by 2137 (47 self)
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We demonstrate that Ethernet LAN traffic is statistically selfsimilar, that none of the commonly used traffic models is able to capture this fractallike behavior, that such behavior has serious implications for the design, control, and analysis of highspeed, cellbased networks, and that aggregating streams of such traffic typically intensifies the selfsimilarity (“burstiness”) instead of smoothing it. Our conclusions are supported by a rigorous statistical analysis of hundreds of millions of high quality Ethernet traffic measurements collected between 1989 and 1992, coupled with a discussion of the underlying mathematical and statistical properties of selfsimilarity and their relationship with actual network behavior. We also present traffic models based on selfsimilar stochastic processes that provide simple, accurate, and realistic descriptions of traffic scenarios expected during BISDN deployment.
On the use of Fractional Brownian Motion in theory of connectionless networks
 IEEE Journal on Selected Areas in Communications
, 1995
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Wavelet Analysis of Long Range Dependent Traffic
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 1998
"... A Wavelet based tool for the analysis of long range dependence is introduced and a related semiparametric estimator of the Hurst parameter. The estimator is shown to be unbiased under very general conditions, and efficient under Gaussian assumptions. It can be implemented very efficiently allowing t ..."
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Cited by 265 (22 self)
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A Wavelet based tool for the analysis of long range dependence is introduced and a related semiparametric estimator of the Hurst parameter. The estimator is shown to be unbiased under very general conditions, and efficient under Gaussian assumptions. It can be implemented very efficiently allowing the direct analysis of very large data sets, and is highly robust against the presence of deterministic trends, as well as allowing their detection and identification. Statistical, computational and numerical comparisons are made against traditional estimators including that of Whittle. The estimator is used to perform a thorough analysis of the long range dependence in Ethernet traffic traces. New features are found with important implications for the choice of valid models for performance evaluation. A study of mono vs multifractality is also performed, and a preliminary study of the stationarity with respect to the Hurst parameter and deterministic trends.
Wavelet Analysis of LongRangeDependent Traffic
, 1998
"... A waveletbased tool for the analysis of longrange dependence and a related semiparametric estimator of the Hurst parameter is introduced. The estimator is shown to be unbiased under very general conditions, and efficient under Gaussian assumptions. It can be implemented very efficiently allowing ..."
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Cited by 146 (1 self)
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A waveletbased tool for the analysis of longrange dependence and a related semiparametric estimator of the Hurst parameter is introduced. The estimator is shown to be unbiased under very general conditions, and efficient under Gaussian assumptions. It can be implemented very efficiently allowing the direct analysis of very large data sets, and is highly robust against the presence of deterministic trends, as well as allowing their detection and identification. Statistical, computational, and numerical comparisons are made against traditional estimators including that of Whittle. The estimator is used to perform a thorough analysis of the longrange dependence in Ethernet traffic traces. New features are found with important implications for the choice of valid models for performance evaluation. A study of mono versus multifractality is also performed, and a preliminary study of the stationarity with respect to the Hurst parameter and deterministic trends.
Analysis of an ATM Buffer with SelfSimilar ("Fractal") Input Traffic
, 1995
"... As ATM highspeed, cellrelay networks will most likely first make their impact as backbones interconnecting enterprise networks consisting of Ethernet and other LANs, their proper design and control is crucial. Recent studies of high quality, high resolution traffic measurements in Bellcore Etherne ..."
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Cited by 99 (1 self)
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As ATM highspeed, cellrelay networks will most likely first make their impact as backbones interconnecting enterprise networks consisting of Ethernet and other LANs, their proper design and control is crucial. Recent studies of high quality, high resolution traffic measurements in Bellcore Ethernets have revealed that this aggregate Ethernet traffic is selfsimilar ("fractal ") in nature, quite different in "burstiness" features from traffic considered and studied up to now. This paper presents an analytical study of an ATM buffer driven with selfsimilar traffic. The probability of buffer occupancy is obtained. It is shown that this probability decreases with the buffer size not exponentially, as in traditionally Markovian traffic models, but algebraically . 1 Introduction Recent studies of highquality, high resolution traffic measurements have revealed a new phenomenon with potentially important ramifications to the modeling, design and control of broadband networks. These includ...
Point Process Approaches for Modeling and Analysis of SelfSimilar Traffic: Part II  Applications
, 1997
"... In previous work [24], Fractal Point Processes (FPPs) have been proposed as novel tools for understanding, modeling and analyzing diverse types of selfsimilar traffic behavior. We apply the FPP models in the context of network traffic modeling and performance analysis. Two qualitatively different f ..."
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Cited by 57 (8 self)
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In previous work [24], Fractal Point Processes (FPPs) have been proposed as novel tools for understanding, modeling and analyzing diverse types of selfsimilar traffic behavior. We apply the FPP models in the context of network traffic modeling and performance analysis. Two qualitatively different fractal data sets (Bellcore Ethernet traces) are characterized by FPP models. Comparison of modeldriven and tracedriven queueing simulation results show that the matched models yield close agreement with the traces over a wide range of system parameters. We also show that under suitable conditions, the FPP models yield Gaussian processes. Queueing simulation shows that the FPP models can be computationally efficient alternatives for generating fractional Gaussian noise processes. Finally, we divide fractal traffic into two types, applicationlevel fractal traffic and networklevel fractal traffic, and argue that each type has radically different implications for the design and control of fut...
Heavy Traffic Analysis of a Storage Model with Long Range Dependent On/Off Sources.
, 1996
"... this paper, we analyze a fluid or storage queueing system with LRD input. Fluid systems have been used before (e.g. Bensaou et al. [2], Guibert [7]) to model bursty traffic fed into ATM multiplexer queues, when considering time scales where the granularity of the ATM cells no longer dominates. The i ..."
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Cited by 35 (6 self)
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this paper, we analyze a fluid or storage queueing system with LRD input. Fluid systems have been used before (e.g. Bensaou et al. [2], Guibert [7]) to model bursty traffic fed into ATM multiplexer queues, when considering time scales where the granularity of the ATM cells no longer dominates. The input sources are assumed to be of On/Off type, that is, with mutually independent, alternating silence periods (no work arriving) and activity periods (work arriving at a constant rate). We consider a superposition of N identical, independent On/Off sources flowing into an infinite reservoir with fixed output rate C. The object of study is the complementary distribution function Q of the stationary queue content. F. Brichet et al. / Heavy traffic analysis with LRD sources 3 If is the mean arrival rate for a single source, we require C ? N for stability. For N sufficiently large, C then exceeds the peak rate of an individual source. If this rate is proportional to C=N , so that it decreases with N , then we are in the realm of "small" sources. The M=G=1 example above represents the limiting form of the alternative assumption, where instantaneous arrivals idealize the case of a capacity significantly smaller than the individual arrival rate. By selecting "heavy" tails (as defined in section 4) for the silence and/or activity periods, the input process becomes long range dependent and the queueing problem is fundamentally nonMarkovian. Despite this, a useful lower bound L to
Point Process Models for SelfSimilar Network Traffic, with Applications
, 1997
"... Selfsimilar processes based on fractal point processes (FPPs) provide natural and attractive network tra#c models. We show that the point process formulation yields a wide range of FPPs which in turn yield a diversity of parsimonious, computationally e#cient, and highly practical asymptotic second ..."
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Cited by 21 (4 self)
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Selfsimilar processes based on fractal point processes (FPPs) provide natural and attractive network tra#c models. We show that the point process formulation yields a wide range of FPPs which in turn yield a diversity of parsimonious, computationally e#cient, and highly practical asymptotic secondorder selfsimilar processes. Using this framework, we show that the relevant secondorder fractal characteristics such as longrange dependence (LRD), slowlydecaying variance, and 1/f noise are completely characterized by three fundamental quantities: mean arrival rate, Hurst parameter, and fractal onset time. Four models are proposed, and the relationship between their model parameters and the three fundamental quantities are analyzed. By successfully applying the proposed models to Bellcore's Ethernet traces, we show that the FPP models prove useful in evaluating and predicting the queueing performance of various types of fractal tra#c sources. Keywords: point process, fractal, selfsimilarity, longrange dependence, tra#c modeling 1 Throughout this paper, selfsimilarity refers to asymptotic secondorder selfsimilarity [4], [13] unless otherwise defined. 1
The mystery of the missing scales: Pitfalls in the use of fractal renewal processes to simulate LRD processes
 IN APPLICATIONS OF HEAVY TAILED DISTRIBUTIONS IN ECONOMICS, ENGINEERING AND STATISTICS, AMERICAN
, 1999
"... It has now been demonstrated in many studies that network traffic exhibits properties consistent with Long Range Dependence (LRD) and selfsimilarity. While theoretical frameworks are currently being developed to estimate the performance of such systems, simulation will remain a valuable tool for va ..."
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Cited by 9 (2 self)
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It has now been demonstrated in many studies that network traffic exhibits properties consistent with Long Range Dependence (LRD) and selfsimilarity. While theoretical frameworks are currently being developed to estimate the performance of such systems, simulation will remain a valuable tool for validating these theoretical models, and providing insight into systems which are too complicated to effectively model. Furthermore, when testing real systems, it is desirable to have traffic sources which are realistic, and hence display selfsimilarity. The Fractal Renewal Process (FRP) and its variants (including On/Off processes and superpositions thereof) have been proposed as models for LRD processes, in particular for network traffic. The FRP is a simple renewal point process with heavytailed interrenewal times. The longrange correlations in the process are directly introduced by the heavy tail of the renewal times. The FRP has the great advantage that the number of computations required to generate a time series is linear and the time series can be generated online, facilitating generation of real traffic. However, there are some problems which arise when using such processes to generate LRD traffic. Most notably undersampling of the heavytailed random variables used to generate FRPs can lead to a truncation of the sampled autocorrelation that is not consistent with LRD. This problem becomes clear when the processes are investigated using the wavelet based methods of Abry and Veitch which segregate behaviour at different scales. This paper will describe the problem of undersampling, and its effects, and methods for avoiding the problem.
Modeling Packet Traffic with Chaotic Maps
, 1994
"... We investigate the application of deterministic chaotic maps to model traffic sources in packet based networks, motivated in part by recent measurement studies which indicate the presence of significant statistical features in packet traffic more characteristic of fractal processes than conventional ..."
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Cited by 5 (1 self)
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We investigate the application of deterministic chaotic maps to model traffic sources in packet based networks, motivated in part by recent measurement studies which indicate the presence of significant statistical features in packet traffic more characteristic of fractal processes than conventional stochastic processes. We outline one approach whereby traffic sources can be modeled by chaotic maps, and illustrate the traffic characteristics that can be generated by analyzing three maps. We show that low order nonlinear maps can capture several of the fractal properties observed in actual data. Finally, we outline a potential performance analysis approach based on chaotic maps that can be used to assess the traffic significance of fractal properties. It is our conclusion that while there are considerable analytical difficulties, chaotic maps may allow accurate, yet concise, models of packet traffic, with some potential for transient and steady state analysis.